SÁRKÖZY'S THEOREM IN VARIOUS FINITE FIELD SETTINGS.

In this paper, we strengthen a result by Green about an analogue of Sárközy's theorem in the setting of polynomial rings F_q[x]. In the integer setting, for a given polynomial F ∈ Z[x] with constant term zero, (a generalization of) Sárközy's theorem gives an upper bound on the maximum size of a subset A ⊂{1, ..., n} that does not contain distinct α₁, α₂ ∈ A satisfying α₁, -- α₂ = F(b) for some b ∈ Z. Green proved an analogous result with much stronger bounds in the setting of subsets A ⊂ F_q[x] of the polynomial ring F_q[x], but this result required the additional condition that the number of roots of the polynomial F ∈ F_q[x] be coprime to q. We generalize Green's result, removing this condition. As an application, we also obtain a version of Sárközy's theorem with similar strong bounds for subsets A ⊂ F_q for q=pⁿ for a fixed prime p and large n. [ABSTRACT FROM AUTHOR]